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Kato-Rellich theorem (Theorem)

Let $ \mathcal{H}$ be a Hilbert space, $ A\colon D(A)\subset \mathcal{H}\to \mathcal{H}$ a self-adjoint operator and $ B\colon D(B)\subset\mathcal{H}\to \mathcal{H}$ a symmetric operator with $ D(A)\subset D(B)$.

We say that $ B$ is $ A$-bounded if there are positive constants $ \alpha,\beta$ such that

$\displaystyle \Vert Bx\Vert\leq \alpha\Vert Ax\Vert+\beta\Vert x\Vert$
for all $ x\in D(A)$, and we say that $ \alpha$ is an $ A$-bound for $ B$.
Theorem (Kato-Rellich)   If $ B$ is $ A$-bounded with $ A$-bound smaller than $ 1$, then $ A+B$ is self-adjoint on $ D(A)$, and essentially self-adjoint on any core of $ A$. Moreover, if $ A$ is bounded below, then so is $ A+B$.



"Kato-Rellich theorem" is owned by Koro.
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Other names:  Rellich-Kato theorem
Also defines:  A-bounded, A-bound
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Cross-references: bounded, core, essentially self-adjoint, self-adjoint, positive, symmetric operator, self-adjoint operator, Hilbert space

This is version 4 of Kato-Rellich theorem, born on 2004-12-11, modified 2006-09-16.
Object id is 6562, canonical name is KatoRellichTheorem.
Accessed 4229 times total.

Classification:
AMS MSC: 47A55 (Operator theory :: General theory of linear operators :: Perturbation theory)
Pending Errata and Addenda
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